4: Lecture Chemical Transport in
the Atmosphere
Suggested Reading: SP Chapter 17
Atmospheric Chemistry
CHEM-5151 / ATOC-5151
Spring 2005
Prof. Brian Toon (PAOS)
The Aerosol Continuity Equation
A. transport
The change in the concentration of a chemical often can be written
as
C
= P LC
t
Here L is the loss rate, P is the production rate, and C is the
species concentration (per unit volume).
1
How do we account for the effects of transport on C?
An observer standing at a
fixed point in space measures
changing concentrations.
The observer must account
for the chemical sources and
sinks as well as for the
motion of the air. This is
called an Eulerian
measurement since it is at a
point.
UC
dd
y
z
Uu C
x
u
Wind direction
UC
dd
y
z
Uu C
u
x
Wind direction
The flux of material into the upwind side of the box is UuCu
2
(particles per cm per sec.)
The total number of particles being added per second is
UuCudydz (particles per second), where dydz is the area of the
open face of the box.
Therefore, considering that material is also leaving the box on
the downwind side, the total amount of material added to the box
per second, divided by the volume of the box so that we have the
particles added cm-3 s-1, is
C
= (U u C u U d C d ) / dx = dUC / dx
t
2
Considering all three directions and adding the
sources and sinks, we arrive at the flux form of the
continuity equation
C (UC) (VC) (WC)
=
+P-LC
t
x
y
z
This is often called the total derivative in m
dC
= P LC
dt
Example of flux based Eulerian transport:
U=10 km/hr
1013 mol cm-3
9x1012 mol cm-3
1 km
L=1013 mol cm-3 hr-1
Assume the wind speed is
constant at 10 km hr-1.
Assume the concentration
declines by 1012 molecules
cm-3 km-1 in the wind
direction.
Assume the concentration
declines by 1013 molecules
cm-3 hr-1 due to a chemical
sink.
The rate of change in the concentration at the fixed downwind
position is
12
13
C
km 9x10 10 mol
13 mol
= 10
(10
)(
)= 0
1
t
hr
km
hr
Hence, in this example the advection by the wind completely
masks the ongoing chemical loss of the material.
3
The Lagrangian form of the continuity equation
U
Ud
An observer moving with the
wind, so that the same air
parcel is always observed,
only has to account for
physical and chemical
changes within the air parcel,
and not for air motions, to
understand how the mixing
ratio varies.
Uu
The Lagrangian form of the continuity equation
Since neither air molecules, nor the species being observed
can be lost from within the parcel, the ratio of the species
concentration to the air density is not changed no matter
how winds distort the volume of the air parcel.
d(C/ )
dt
= (P LC)/
4
A view of aerosol transport,
There were large fires in Russia prior to this time period,
and dust storms in Africa. Can you tell the source and sink regions just
by glancing at the distributions and knowing the winds?
Transport of
forest fire smoke
in July 2002 from
Seawifs
5
Polar nucleation
layer
Mixing ratios can identify
source and sink regions
Tropical
nucleation
The diffusion approximation in the continuity
equation
Brownian Diffusion occurs due to the relative random
motions of air molecules.
The Brownian diffusion equation can be derived from THE
KINETIC THEORY OF GASES.
In fluid mechanics turbulent motions can be approximated using
equations similar to those from Brownian diffusion.
Therefore, atmospheric chemists have developed an approximate
theory which is referred to as eddy diffusion.
Eddy diffusion is not real, often is misleading, and usually is not
used to represent turbulence, but instead the large scale
circulation. Still it is widely used.
6
The diffusive flux, in analogy to thermodynamics is
C
( )
K X
x
There is no diffusive flux if the
mixing ratio is independent of
location.
The diffusive flux is often
referred to as being "down the
gradient", which means diffusion
causes a positive flux in the
direction of decreasing mixing
ratio.
Hence diffusion produces a
uniform mixing ratio by
transporting material from regions
where the mixing ratio is high into
regions where the mixing ratio is
low
The diffusion coefficient
100
Altitude, km
80
Brownian
60
40
K -1/2
z
Brasseur and Solomon
Diffusion Coefficient
20
0
10
2
10
3
4
5
10
10
Diffusion Coefficient, cm
2
10
s -1
6
10
7
A typical eddy diffusion coefficient used in one -dimensional models of
the atmosphere.
The Brownian diffusion coefficient is much smaller than the eddy
diffusion coefficient below 100 km
7
Three views of transport
Example descent into the polar vortex. During polar night air
descends from the mesopause into the lower stratosphere. How
can we think about this process?
1. The Eulerian view
Descending air
29 km
27 km
Polar vortex
WCtop
27 km
25 km
South pole
WCbottom
WCtop> WCbottom
Three views of transport
2. the Lagrangian View
Z=50 km
P=0.8mb
ar
dP=1mbar
dz=10km
dP=1mbar
dz=1.5 km
Z=37 km
P=4mbar
Z=25 km
2P=5mbar
dP=1mbar
dz=300m
8
Three views of transport
50km
3. The diffusion view
50km
25km
25km
Ozone Mixing ratio
Aerosol Mixing ratio
50 km
25 km
The ADVANTAGE OF THE DIFFUSION EQUATION IS
THAT IT CAN BE SOLVED RELATIVELY EASILY.
CONSIDER THE FOLLOWING SIMPLE TRANSPORT AND
CHEMISTRY PROBLEM
1. ASSUME THAT THE CONCENTRATION OF A MATERIAL
IS HELD CONSTANT AT THE SURFACE
2. ASSUME THAT VERTICAL TRANSPORT BY EDDY
DIFFUSION ACTS AGAINST A CONSTANT, ALTITUDE
INDEPENDENT CHEMICAL LOSS RATE
3. THE STEADY STATE EQUATION TO BE SOLVED IS
( K z
(C / )
)
z
= LC
z
9
The LOSS RATE IS THE INVERSE OF THE CHEMICAL
LIFETIME
C = 1/ L
The AIR DENSITY IS A SIMPLE FUNCTION OF ALTITUDE IF
THE ATMOSPHERE IS ISOTHERMAL
= o exp(
z
)
H
SO WE CAN REWRITE THE EQUATION AS
2 C 1 C C
+
= 0.
z 2 H z K
THE SOLUTION TO THE EQUATION IS
C
=
C0
0
2
exp
z
H
( 0.25 +
0.5)
H
K c
The chemical time constant appears in a ratio with another
time constant for vertical transport.
H2
=
Kz
d
10
The dynamical lifetime d= H2/Kz for several fixed
values of diffusion coefficient for and a typical
altitude dependent diffusion coefficient
100
80
10
5
3
cm
2
s
-1
10
4
cm
2
s
-1
10
3
cm
2
s
-1
60
40
Variable K
z
20
0
0.01
0.1
1
Dynamics time constant,yrs
10
10
2
The vertical variation of the mixing ratio (assuming
a unit mixing ratio at the surface for simplicity) for
various values of the ratio of the chemical lifetime
to the dynamical lifetime.
100
100
80
10
1
60
40
0.1
20
0.01
0
-4
10
10
-3
-2
10
mixing ratio
10
-1
10
0
When the chemical lifetime is
100 times larger than the
dynamical lifetime, materials
will have an almost constant
mixing ratio to nearly 100 km
altitude.
However, when the chemical
lifetime is 1% of the
dynamical lifetime the mixing
ratio falls very rapidly in the
troposphere.
11
Numerical and analytical solutions of the diffusion
equation.
100
/ =10
c
/ =10
c
Altitude, km
80
d
numerical
d
analytic
Exponential
lifetime
Variable K
z
60
40
20
Exponential lifetime
and variable K
z
0
10
-1
10
0
mixing ratio
1. Solid red-chemical lifetime is ten times the dynamical lifetime.
2. Dotted black the chemical lifetime is held constant, but the transport is
done with the vertically varying diffusion coefficient .
3. Green- constant diffusion coefficient of 104 cm2 s-1, but the chemical
lifetime decreases exponentially with altitude using a scale height of 4H
Where H=7 km.
4.Dashed red- the diffusion coefficient varies with altitude, and the
chemical lifetime decreases exponentially with altitude.
Atmospheric observations
Material
H2O
CH4
COS
SO2
N2O
CFC-11
CFC-12
CH3Cl
NaCl
Lifetimes of some interesting materials
_____________________________________________________________________________
Mb, Abundance (Tg)
Pb, Source
tc, Lifetime
(Tg/yr)
(yr)
1.3x107
5x108
0.025
5x103
515
10
5.2
1.2
4.3
0.6-0.9
200
.003-.005
2.5x103
12-21
120
6.2
0.25
50
10.3
0.37
102
5
3.5
1.5
3.6
1300
0.003
12
METEOROLOGICAL TRACERS
IT IS VERY USEFUL TO HAV E
METEOROLOGICAL TRACERS SO THAT THE
PATHS ALONG WHICH AIR PARCELS MOV E
CAN BE IDENT IFIED
CON SIDER THE FIRST LAW OF
THERMOD YNA MICS REWRITTEN
IDEAL GAS LAW
R dlnp
1 dQ
dlnT
= cp
M dt
T dt
dt
WITH THE
IF WE CON SIDER AD IABATIC TRAN SPORT IN
WHICH NO HEAT ING OCCURS THEN WE CAN
IN TEGRAT E THE TEMPERATURE OV ER
ALTITUD E AN D GET
dlnT
T
1000mbars
=
P
R
dlnP
Mc p
YIELDIN G
1000mbars
= T(
p
)
R
Mc p
IS CALLED THE POTENT IAL TEMPERAT URE.
IT IS THE TEMPERAT URE THAT AN AIR
PARCEL WOULD HAV E IF IT WERE TAKEN
AD IABA TICALLY TO A PRESSU RE OF 1000
MBARS.
IS A CON SERVED TRACER
TAKING
THE
LOGAR ITHM
OF
,
DIFFERENTIATING WITH RESPECT TO TIME,
AN D
USING
THE
FIRST
LAW
OF
THERMO D YNA MICS
YIELDS
THE
LAGRANG IAN
FORM OF THE CONT INU IT Y
EQUAT ION
FOR
THE
POTENTIA L
TEMPERAT URE:
dQ
d
=
c p T dt .
dt
IS CON SERVED BY AIR PARCELS WHICH DO
NO T EXPERIENCE AN Y EXTERNA L HEAT ING.
OV ER SHORT TIME SCALES, OFTEN SEVERAL
DAYS, EXTERNA L HEAT ING DUE TO
RAD IAT ION IS USU ALLY SMA LL. SO AIR
PARCELS WHICH DO NO T PASS THROUGH
CLOUDS, APPROXIMAT ELY MOV E ALONG
SURFACES OF CON STAN T POTENT IAL
TEMPERAT URE. SUCH SURFACES CAN BE
FOUN D FROM ANA LYSES OF THERMA L
STRUCTURE. WIN DS ON THESE SURFACES
ALLOW THE TRAJECTO RIES OF AIR PARCELS
TO BE CALCULATED. THESE TRAJECTO RIES
ALLOW STUDIES OF LAGRANGIAN
CHEMISTRY. SO THE RECOGNI TION OF
CON STAN T POTENT IAL TEMPERATURE
SURFACES CONV ERTS THE THREEDIMENSION AL CHEMICAL TRAN SPORT
13
M EASU RES THE STA BILITY OF THE
ATMOSPHERE
FROM THE FIRST LAW OF THERMOD YNA MICS
AN D THE CONT INU ITY EQUAT ION FOR WE
GET
RT
d
dQ = C p dT
dP = TC p
MP
USING THE HYDR OSTAT IC EQUAT ION
DIVID IN G BY CPTdz YIELDS
1 d 1 dT RT( gdz 1 dT
g
=
+
=
T dz
dz
T dz RT
Cp
C p Tdz
M
M
So
d
dz
d
AN D
= 1
d
T
(
> 0 stable
= 0 neutral
dz
d
< 0 unstable
dz
14
1
POTENT IAL VO RTICITY
ANO THER USEFUL METEOROLOGICAL
TRACER IS POTENT IAL VO RTICITY. IT IS
ANA LOGOU S TO ANG ULAR MOMENT UM
J= R2
2>1
THE POTENT IAL VO RTICITY PV OBEYS
d[ g( + f ) ]
d[PV ]
p
=
=0,
dt
dt
IF THE FLOW IS AD IABATIC AN D
FRICTIONLESS ( IE DIFFUSION ISNT
IMPORTAN T DUE TO SMA LL SCALE
TRAN SPORT) . SO PV AN D A RE
CON SERVED UN DER THE SAM E
COND ITION S.
PV HAS UN ITS OF K CM 2 G-1 S -1
PV = g( + f )
p
f = 2 sin
IS A MEASU RE OF THE ROTA TION OF AN AIR
PARCEL DUE TO ITS LOCA TION ON THE
EARTH, IT HAS S -1 U N ITS.
, IS THE VERTICAL COMPON ENT OF THE
RELATIVE VO RTICITY OF THE FLUID , A
MEASU RE OF THE MICROSCOPIC TEN DENCY
OF THE FLUID TO ROTA TE DUE TO WIN DS
AN D HAS UN ITS OF S -1
=
lim V dl
A 0
A
or
x y = U x + (V +
V
x
x)y (U +
U
y
y ) x Vy
or
=
V U
x y
__ y)
(U+
U
y
V
y
(V+
V
__ x)
x
U
x
15
THE FINA L PART OF THE DEFINITION OF
POTENT IAL VO RTICITY IS THE VERTICAL
GRAD IENT OF . THIS AS A MEASU RE OF THE
DEPTH OF THE FLUID .
EXAM PLE:
CON SIDER A UN IFORM ( NO GRAD IENT S IN THE
HORIZONTA L DIRECTION S) WESTERLY FLOW
OF AIR, OV ER A CHAIN OF MOUNTA INS IN
THE NO RTHERN HEMISPHERE.
SINCE THE AIR FLOW IS ASSUMED TO BE
UN IFORM IN ITIALLY IT HAS NO RELATIVE
VO RTICITY. PV WILL BE POSITIVE DUE TO
THE CORIOLIS TERM.
THE SURFACE AT THE BASE OF THE FLOW
MUST RISE AS THE AIR MOV ES OV ER THE
MOUNTA INS SO THE DEPTH OF THE FLUID
IS DECREASED, AN D THE GRAD IENT OF
WITH PRESSU RE IS IN CREASED.
THUS THE
CHANG E IN THE TERM IN PV IN VOLV ING
WILL BE IN THE SENS E TO IN CREASE PV.
TO CON SERVE PV, THE RELATIVE VO RTICITY
OF THE FLUID MUST BECOM E NEGATIVE SO
THAT IT CAN REMOV E SOME OF THE PV
DUE TO THE CORIOLIS TERM.
16
FOR THE RELATIVE VO RTICITY TO BECOME
NEGATIVE THE AIR MUST TURN TOWARD
THE SOU TH .
AS THE AIR LEAV ES THE MOUNTA INS THE
GRAD IENT OF W ILL DECREASE AS THE
DEPTH OF THE AIR PARCEL IN CREASES.
THEN THE AIR WILL SWIN G BACK TOWARD
THE NO RTH.
HENCE CON SERVA TION OF PV REQU IRES AN
OSCILLATORY MOTION BE IN DUCED AS AIR
FLOWS OV ER A MOUNTA IN RANG E. SUCH
OSCILLATION S ARE SEEN ON DAILY WEATHER
MA PS WHERE AIR FLOWS OV ER EXTENDED
MOUNTA IN RANG ES, SUCH AS THE ROCKIES.
17
Tracer/tracer plots are useful
18
19